Elsevier-funded NY Congresswoman Carolyn Maloney Wants to Deny Americans Access to Taxpayer Funded Research

Fraser has this warning:

The failure to make true assertions with a promised reliability can be extreme with the Bayes use of mathematical priors (Stainforth et al., 2007; Heinrich, 2006). The claim of a probability status for a statement that can fail to be approximate confidence is misrepresentation. In other areas of science such false claims would be treated seriously.

The complaint about coverage of Bayesian confidence regions arises often, I think, because there is a very ingrained notion that correct frequentist coverage is a most desirable quality; that frequentist coverage is a literal statement about a natural phenomenon. Fraser goes further to say that confidence regions with incorrect coverage are misrepresentative, perhaps even fraudulent (i.e., a thing to be 'treated seriously')!

Of course, frequentist coverage is almost never a literal statement about a natural phenomenon, because statistical models almost never fully reflect the truth. In the sentiment of G.E.P. Box, all reported confidence levels are wrong, but some are useful.

More importantly, the criticism of approximate frequentist coverage is readily dismissed from a Bayesian perspective. In response, Christian Robert had this final comment:

Bayesian credible intervals are not frequentist confidence intervals and thus do not derive their optimality from providing an exact frequentist coverage.

On a side note, Fraser's statement above seems to neglect that frequentist probability is different from Bayesian probability in the context of confidence regions! I wrote on a related topic a few weeks ago: Bayesian vs. Frequentist Intervals: Which are more natural to scientists? Unfortunately though, my understanding is upset again due to Robert's reference to Jaynes: "there is only one kind of probability". How can this be true? Aren't Bayesians clear that the posterior distribution on a parameter is not to be interpreted in a frequentist way? And aren't frequentists clear that confidence regions not to be interpreted in a subjective way?

Mortgage rates are low, considering historical rates for the last 50 years. It may be timely to consider a mortgage refinance. The image above links to a simple tool for exploring mortgage refinance, built using rapache and the yet-to-be-archived yarr package for R. Hence, there are now two mortgage-related calculators on this site:

• MortCalc: A tool to explore new mortgages.
• RefiCalc: A tool to explore mortgage refinance.

While I believe both calculators to be accurate, I make no guarantees. You are welcome to browse the source code (single file, mixed R and HTML/CSS) for either calculator: MortCalc.yarr, RefiCalc.yarr.

The rule changes are summarized as follows:

• cost to play increased to $2 from 1$
• jackpot starts at $40M instead of $20M
• second prize starts at $1M instead of $0.2M
• number of red balls (powerballs) reduced to 34 from 39
• prize for matching the red ball alone increased to $4 from $3

The MUSL is touting the changes using the slogan "MORE, BIGGER, BETTER: more millionaires, bigger starting jackpots, and better overall odds." However, as I demonstrate below, the average loss incurred by playing Powerball will increase by more than two times, from $0.49 to $1.15. That is, under the new rules, Powerball players can expect to lose $1.15 for every $2 ticket purchased. In the long run, this is equivalent to worse than exchanging a $2 bill for a $1 bill every time Powerball is played.

Note that these calculations are approximate because the jackpot is estimated using historical jackpots, and very generous because taxes are not deducted from winnings.

The following R code computes the expected winnings (loss) under the old and new rules. You download the script or source the file from within R like so:

R> source("http://biostatmatt.com/R/powerball.R")

# jackpot amounts (as of 12/16/2011)
# http://www.lottostrategies.com/script/jackpot_history/draw_date/101
# http://www.usamega.com/powerball-jackpot.asp
jackpots <- read.csv('http://biostatmatt.com/csv/jackpot.csv')
average_jackpot <- mean(jackpots$PrizeMillions)

# for convenience
ch <- choose

# old rules
# choose 5 from 59 white balls, 1 from 39 red balls

# probabilities of winning
powerball_old <- function(w, r)
    ch(5,w)*ch(54,5-w)/ch(59,5)*ch(1,r)*ch(38,1-r)/ch(39,1)

# 9 ways to win
p_old <- vector("numeric", length=9)
p_old[1] <- powerball_old(5, 1) # five white + powerball_old
p_old[2] <- powerball_old(5, 0) # five white
p_old[3] <- powerball_old(4, 1) # four white + powerball_old
p_old[4] <- powerball_old(4, 0) # four white
p_old[5] <- powerball_old(3, 1) # three white + powerball_old
p_old[6] <- powerball_old(3, 0) # three white
p_old[7] <- powerball_old(2, 1) # two white + powerball_old
p_old[8] <- powerball_old(1, 1) # one white + powerball_old
p_old[9] <- powerball_old(0, 1) # powerball_old

# winnings
w_old <- vector("numeric", length=9)
w_old[1] <- 1000000 * average_jackpot
w_old[2] <- 200000
w_old[3] <-  10000
w_old[4] <-    100
w_old[5] <-    100
w_old[6] <-      7
w_old[7] <-      7
w_old[8] <-      4
w_old[9] <-      3

# expected winnings (loss)
# cost to play is $1
expected_winnings_old_rules <- -1 * (1 - sum(p_old)) + sum(p_old * (w_old - 1))

# new rules
# choose 5 from 59 white balls, 1 from 35 red balls
# probabilities of winning powerball
powerball_new <- function(w, r)
    ch(5,w)*ch(54,5-w)/ch(59,5)*ch(1,r)*ch(34,1-r)/ch(35,1)

# 9 ways to win
p_new <- vector("numeric", length=9)
p_new[1] <- powerball_new(5, 1) # five white + powerball_new
p_new[2] <- powerball_new(5, 0) # five white
p_new[3] <- powerball_new(4, 1) # four white + powerball_new
p_new[4] <- powerball_new(4, 0) # four white
p_new[5] <- powerball_new(3, 1) # three white + powerball_new
p_new[6] <- powerball_new(3, 0) # three white
p_new[7] <- powerball_new(2, 1) # two white + powerball_new
p_new[8] <- powerball_new(1, 1) # one white + powerball_new
p_new[9] <- powerball_new(0, 1) # powerball_new

# winnings
# jackpots start $20M larger than before
# second prize is $1M instead of $0.2M
# prize for powerball only is $4 instead of $3
w_new <- vector("numeric", length=9)
w_new[1] <- 1000000 * (average_jackpot + 20)
w_new[2] <- 1000000
w_new[3] <-  10000
w_new[4] <-    100
w_new[5] <-    100
w_new[6] <-      7
w_new[7] <-      7
w_new[8] <-      4
w_new[9] <-      4

# expected winnings (loss)
# cose to play is $2 instead of $1
expected_winnings_new_rules <- -2 * (1 - sum(p_new)) + sum(p_new * (w_new - 2))

I found this picture apropos (from http://www.toxel.com/inspiration/2009/09/18/12-creative-toilet-paper-designs/):

If you can't pay your taxes, borrow the money!

[Pub 17, pp. 17, 2011] If you cannot pay the full amount [of tax] due with your return, you can ask to make monthly installment payments for the full or a partial amount. However, you will be charged interest and may be charged a late payment penalty... But, before requesting an installment agreement, you should consider other less costly alternatives, such as a bank loan. [emphasis mine]

This might interest Warren Buffet and other billionaires who want to contribute more:

[Pub 17, pp. 17, 2011] You can make a contribution (gift) to reduce debt held by the public. If you wish to do so, make a separate check payable to “Bureau of the Public Debt.” Send your check to:

Bureau of the Public Debt
ATTN: Department G
P.O. Box 2188
Parkersburg, WV 26106-2188.

You can deduct this gift as a charitable contribution on next year’s tax return...

I visited the Bureau of the Public Debt website and found the following statistics (in U.S. dollars):

Current	Debt Held by the Public	Intragovernmental Holdings	Total Public Debt Outstanding
12/08/2011	10,392,391,725,777.77	4,658,769,264,160.46	15,051,160,989,938.23

It turns out that gifts are sometimes made (source):

In the U.S., married taxpayers should figure tax three times:

[Pub 17, pp. 24, 2011] You will generally pay more combined tax on separate returns than you would on a joint return... However, ... you should figure your tax both ways (on a joint return and on separate returns). This way you can make sure you are using the filing status that results in the lowest combined tax.

This is silly, isn't there a better way than "guess-and-check"?

Don't forget to report bribes, kickbacks, stolen property, and income from other illegal activities:

[Pub 17, pp. 94, 2011] Bribes. If you receive a bribe, include it in your income.Illegal activities. Income from illegal activities, such as money from dealing illegal drugs, must be included in your income... Kickbacks. You must include kickbacks, side commissions, push money, or similar payments you receive in your income... Stolen property. If you steal property, you must report its fair market value in your income in the year you steal it unless in the same year, you return it to its rightful owner.

Don't give money to the damn commies:

[Pub 17, pp. 166, 2011] You cannot deduct contributions to organizations that are not qualified to receive tax-deductible contributions, including ... communist organizations.

Also under the list of things not deductable:

[Pub 17, pp. 207, 2011] Wristwatches You cannot deduct the cost of a wristwatch, even if there is a job requirement that you know the correct time to properly perform your duties.

My mother visited a few weeks ago and left behind the carnage above. She claims that an overhead light fell into the sink 'all by itself'.

Unfortunately, the news article chose one of the least convincing graphics to summarize this finding, which is reproduced above (from screen capture). In the report, the principal comparison is drawn between 401(k) participants who accept "help" versus those who do not. However, from the report:

In this report, the majority of Help Participants had risk levels in the 10% to 18% range [footnote 15]. We use this range as our proxy for "appropriate" risk levels, which are consistent with diversified portfolios combining equity and fixed income holdings.

[footnote 15] This range represents approximately 85% of participants using Help.

Rather than select the range 10% to 18% directly (as the report leads us to believe by relegating the other pertinent information into a footnote), I suspect that the authors simply selected the range corresponding to 85% of participants "using Help". Of course, such a "proxy for 'appropriate' risk levels" would be completely biased and self-serving! Indeed, the report authors might have selected a range corresponding to 90% of participants "using Help". If the "appropriate" risk range had some external validity, why would the authors hide (in a footnote) the fact that 85% of participants who used "help" had risk within this range?

The Polya urn is a heuristic associated with Dirichlet process mixtures. We present the scheme in a modified format, using balloons instead of balls, where the probability of drawing a balloon from the urn is proportional to its volume. Balloons are preferred because their volume may be adjusted by fractional amounts, whereas a ball count may be adjusted only in whole amounts.

The Polya urn initially contains n uniquely colored baloons, each filled with the same volume of air. At each draw, a single baloon is selected at random from the collection of balloons within the urn, and its color is recorded. If the recorded color had been observed in previous draws, the baloon is inflated by an amount equal to its original volume. Finally, the balloon is returned to the urn. Hence, a balloon drawn from the Polya urn is more likely be observed in subsequent draws. ‘The rich get richer’ is a fitting mnemonic for the Polya urn scheme.

Proponents of Bayesian statistical inference argue that Bayesian credible intervals are more intuitive than the frequentist confidence intervals, because the Bayesian inference is a probability statement about a parameter. A frequentist uses the term 'confidence' because their intervals are based on the probability that a random interval includes the value of a parameter.

A non-statistician scientist might initially agree that the Bayesian interval is more natural because it reads as a probability statement. But, while scientists do often think and behave (perhaps subconsciously) in a Bayesian fashion (i.e., update their prior beliefs using evidence from current experiments), their conscious notion of probability is often more aligned with the frequentist (roughly, the relative frequency of an event in repeated experiments).

For each type of interval, the 'parameter' most often refers to a fixed population quantity used to index a family of parametric models. Hence, when the scientist is reminded of this, the conflict is apparent between the Bayesian interval interpretation and the frequentist notion of probability. The scientist may ask: 'Since I have assumed that a parameter has fixed value, how can I claim that its value will lie in an interval with some frequency in repeated experiments?'

Of course, the above is a misinterpretation (as was pointed out in a previous discussion), because the Bayesian idea of probability is different than the frequentist idea. The scientist may later learn that for Bayesians, 'probability' refers to ones subjective belief about the fixed value of a parameter, but that this belief may be modified by new evidence. The fully informed scientist, despite any subconscious Bayesian tendencies, will often reject the Bayesian notion of probability in favor of the more 'objective' frequentist probability.

So, how should a Bayesian argue more convincingly?

I suppose the title of this post might have been "Bayesian vs. Frequentist Probabilities: ..." I would be surprised if there were not a literature related to this question that I have neglected. But, isn't a blog a reasonable forum to express ideas without the need for exhaustive research to ensure someone else hasn't had the same idea?